Two-dimensional Local Ring Research Articles

Class field theory, Hasse principles and Picard–Brauer duality for two-dimensional local rings

Class field theory, Hasse principles and Picard–Brauer duality for two-dimensional local rings

Arithmetic duality for two-dimensional local rings with perfect residue field

We give a refinement of Saito's arithmetic duality for two-dimensional local rings by giving algebraic group structures for arithmetic cohomology groups.

Bounds on the minimal number of generators of the dual module

Let [Formula: see text] be a Cohen–Macaulay local ring. Let [Formula: see text] be a finitely generated [Formula: see text]-module and let [Formula: see text] denote the [Formula: see text]-dual of [Formula: see text]. Furthermore, if [Formula: see text] is a maximal Cohen–Macaulay [Formula: see text]-module, then we prove that [Formula: see text], where [Formula: see text] is the cardinality of a minimal generating set of [Formula: see text] as an [Formula: see text]-module and [Formula: see text] is the multiplicity of the local ring [Formula: see text]. Furthermore, if [Formula: see text] is a reflexive [Formula: see text]-module then [Formula: see text]. As an application, we study the bound on the minimal number of generators of specific modules over two-dimensional normal local rings. We also mention some relevant examples.

NORMAL HILBERT COEFFICIENTS AND ELLIPTIC IDEALS IN NORMAL TWO-DIMENSIONAL SINGULARITIES

AbstractLet $(A,\mathfrak m)$ be an excellent two-dimensional normal local domain. In this paper, we study the elliptic and the strongly elliptic ideals of A with the aim to characterize elliptic and strongly elliptic singularities, according to the definitions given by Wagreich and Yau. In analogy with the rational singularities, in the main result, we characterize a strongly elliptic singularity in terms of the normal Hilbert coefficients of the integrally closed $\mathfrak m$ -primary ideals of A. Unlike $p_g$ -ideals, elliptic ideals and strongly elliptic ideals are not necessarily normal and necessary, and sufficient conditions for being normal are given. In the last section, we discuss the existence (and the effective construction) of strongly elliptic ideals in any two-dimensional normal local ring.

Gröbner cells of punctual Hilbert schemes in dimension two

We begin with a comprehensive discussion of the punctual Hilbert scheme of the regular two-dimensional local ring in terms of the Gröbner cells. These schemes are the most degenerate fibers of the Grothendieck-Deligne norm map (the Hilbert-Chow morphism), playing an important role in the study of Hilbert schemes of smooth surfaces. They are generally singular, but their Gröbner cells are affine spaces; they admit an explicit parametrization due to Conca and Valla. We use this to obtain the Gröbner decomposition of compactified Jacobians of plane curve singularities, which is non-trivial even for the generalized Jacobians (principal ideals only). One of the applications is the topological invariance of certain variants of compactified Jacobians and the corresponding motivic superpolynomials for analytic deformations of quasi-homogeneous plane curve singularities and some similar families.

The normal reduction number of two-dimensional cone-like singularities

Let $(A, \mathfrak m)$ be a normal two-dimensional local ring and $I$ an $\mathfrak m$-primary integrally closed ideal with a minimal reduction $Q$. Then we calculate the numbers: $\mathrm{nr}(I) = \min\{n \;|\; \overline{I^{n+1}} = Q\overline{I^n}\}, \quad \bar{r}(I) = \min\{n \;|\; \overline{I^{N+1}} = Q\overline{I^N}, \forall N\ge n\}$, $\mathrm{nr}(A)$, and $\bar{r}(A)$, where $\mathrm{nr}(A)$ (resp. $\bar{r}(A)$) is the maximum of $\mathrm{nr}(I)$ (resp. $\bar{r}(I)$) for all $\mathfrak m$-primary integrally closed ideals $I\subset A$. Then we have that $\bar{r}(A) \le p_g(A) + 1$, where $p_g(A)$ is the geometric genus of $A$. In this paper, we give an upper bound of $\bar{r}(A)$ when $A$ is a cone-like singularity (which has a minimal resolution whose exceptional set is a single smooth curve) and show, in particular, if $A$ is a hypersurface singularity defined by a homogeneous polynomial of degree $d$, then $\bar{r}(A)= \mathrm{nr}(\mathfrak m) = d-1$. Also we give an example of $A$ and $I$ so that $\mathrm{nr}(I) = 1$ but $\bar{r}(I)= \bar{r}(A) = p_g(A) +1=g+1$ for every integer $g \ge 2$.

A note on Gersten’s conjecture for étale cohomology over two-dimensional henselian regular local rings

We show the Gersten's conjecture for \'etale cohomology over two dimensional henselian regular local rings without assuming equi-characteristic. As application, we obtain the local-global principle for Galois cohomology over mixed characteristic two-dimensional henselian local rings.

Normal Reduction Numbers for Normal Surface Singularities with Application to Elliptic Singularities of Brieskorn Type

In this paper, we give a formula for normal reduction number of an integrally closed $\mathfrak m$ -primary ideal of a two-dimensional normal local ring $(A,\mathfrak m)$ in terms of the geometric genus pg(A) of A. Also, we compute the normal reduction number of the maximal ideal of Brieskorn hypersurfaces. As an application, we give a short proof of a classification of Brieskorn hypersurfaces having elliptic singularities.

Finite generation of extensions of associated graded rings along a valuation

In this paper we consider the question of when the associated graded ring along a valuation, gr ν ∗ ( S ) , is a finite gr ν ∗ ( R ) -module, where S is a normal local ring which lies over a normal local ring R and ν ∗ is a valuation of the quotient field of S which dominates S. We begin by discussing some examples and results, allowing us to refine the conditions under which finite generation can hold. We must impose the condition that the extension of valuations is defectless and performs a birational extension of R along the valuation to obtain finite generation (replacing S with the local ring of the quotient field of S determined by the valuation which lies over the extension of R). With these assumptions, we have that finite generation holds, when R is a two-dimensional excellent local ring. Our main result (in Theorem 1.5) is to show that for an arbitrary valuation in an algebraic function field over an arbitrary field of characteristic zero, after a birational extension along the valuation, we always have finite generation (all finite extensions of valued fields are defectless in characteristic zero). This generalizes an earlier result, in [Cutkosky, Math. Proc. Cambridge Soc. 106 (2016) 233–255], showing that finite generation holds (after a birational extension) with the additional assumptions that ν has rank 1 and has an algebraically closed residue field. There are essential difficulties in removing these assumptions, which are addressed in the proof in this paper. We obtain general results for unramified extensions of excellent local rings in Proposition 1.7, showing that after blowing up, the extension of associated graded rings is finitely generated of an extremely simple form. This proposition plays an essential role in the Proof of Theorem 1.5.

Effective computation of base points of ideals in two-dimensional local rings

We provide an algorithm that allows to describe the minimal log-resolution of an ideal in a smooth complex surface from the minimal log-resolution of its generators. In order to make this algorithm effective we present a modified version of the Newton–Puiseux algorithm that computes the Puiseux decomposition of a product of not necessarily reduced or irreducible elements together with their algebraic multiplicity in each factor.

Poincaré series of multiplier ideals in two-dimensional local rings with rational singularities

We study the multiplicity of the jumping numbers of an m-primary ideal a in a two-dimensional local ring with a rational singularity. The formula we provide for the multiplicities leads to a very simple and efficient method to detect whether a given rational number is a jumping number. We also give an explicit description of the Poincaré series of multiplier ideals associated to a proving, in particular, that it is a rational function.

A Cohomological Hasse Principle Over Two-dimensional Local Rings

Let $K$ be the fraction field of a two-dimensional henselian, excellent, equi-characteristic local domain. We prove a local-global principle for Galois cohomology with finite coefficients over $K$. We use classical machinery from \'etale cohomology theory, drawing upon an idea in Saito's work on two-dimensional local class field theory. This approach works equally well over the function field of a curve over an equi-characteristic henselian discrete valuation field, thereby giving a different proof of (a slightly generalized version of) a recent result of Harbater, Hartmann and Krashen. We also present two applications. One is the Hasse principle for torsors under quasi-split semisimple simply connected groups without $E_8$ factor. The other gives an explicit upper bound for the Pythagoras number of a Laurent series field in three variables. This bound is sharper than earlier estimates.

Multiplier ideals in two-dimensional local rings with rational singularities

Multiplier ideals in two-dimensional local rings with rational singularities

Equisingularity classes of birational projections of normal singularities to a plane

Given a birational normal extension O of a two-dimensional local regular ring ( R , m ) , we describe all the equisingularity types of the complete m -primary ideals J in R whose blowing-up X = Bl J ( R ) has some point Q whose local ring O X , Q is analytically isomorphic to O .

An avoidance principle with an application to the asymptotic behaviour of graded local cohomology

We present an avoidance principle for certain graded rings. As an application we fill a gap in the proof of a result of Brodmann, Rohrer and Sazeedeh about the antipolynomiality of the Hilbert–Samuel multiplicity of the graded components of the local cohomology modules of a finitely generated module over a Noetherian homogeneous ring with two-dimensional local base ring.

Sums of squares of linear forms

Let k be a real field. We show that every non-negative homogeneous quadratic polynomial f (x(1),..., x(n)) with coefficients in the polynomial ring k[t] is a sum of 2n center dot tau(k) squares of linear forms, where tau(k) is the supremum of the levels of the finite non-real field extensions of k. From this result we deduce bounds for the Pythagoras numbers of affine curves over fields, and of excellent two-dimensional local henselian rings.

An explicit formula for the Hilbert symbol for Honda groups in a multidimensional local field

Based on the pairing on Cartier curves explicitly constructed in the previous paper of the authors, an explicit formula for the Hilbert symbol is constructed in a multidimensional local field of characteristic zero with residue field of positive characteristic on the formal module of a one-dimensional Honda formal group. In the proof a Shafarevich basis on the formal module is constructed, and so-called integer -modules in two-dimensional local rings of a special form ( -rings) are studied.

Rees Algebras of F-regular Type

We study the F-regularity of Rees algebras R(I)=A[It] in terms of the global F-regularity of the blowing-up X=ProjR(I) of SpecA. As it reads, global F-regularity is a global analog of strong F-regularity defined via splitting of Frobenius maps in prime characteristic, and these notions are extended to characteristic zero by reduction modulo p⪢0. We study in detail the case where (A,m) is a two-dimensional local ring and I is an m-primary ideal. In characteristic zero, the condition for R(I) to have F-regular type is described in terms of the dual graph of a resolution X̃ on which IOX̃ is invertible. We also prove some miscellaneous results concerning singularities of Rees algebras and extended Rees algebras of higher dimension.

Hilbert-Kunz Multiplicity of Two-Dimensional Local Rings

We study the behavior of Hilbert-Kunz multiplicity for powers of an ideal, especially the case of stable ideals and ideals in local rings of dimension 2. We can characterize regular local rings by certain equality between Hilbert-Kunz multiplicity and usual multiplicity.We show that rings with “minimal” Hilbert-Kunz multiplicity relative to usual multiplicity are “Veronese subrings” in dimension 2.

A Note on Class Field Theory for Two-Dimensional Local Rings

The class field theory for the fraction field of a two-dimensional complete normal local ring with finite residue field is established by S. Saito. In this paper, we investigate the index of the norm group in the K 2-idele class group for a finite Abelian extension of such fields and deduce that the existence theorem does not hold for almost fields in this case.